No, they don't.
Now, policy makers and media often highlight, and some programs are based on, the “core” CPI which excludes food and energy [0], rather than the full CPI, but since the data is collected by sector anyway, it's virtually zero additional effort to calculate the core CPI in addition to the full CPI; they are just two different aggregates drawn from the same base data.
The full CPI doesn't exclude “a lion share of basic living necessities” (it excludes taxes unrelated to consumer goods purchases, and that's about it.) [1]
[0] in principal, because of high short-term volatility of those components.
Well, yes. This is how it is
The point where the dodgy part is for example how they do weighting for things like housing rents, their criteria of non-comparable replacements, idea of product classes (an apple is not an inferior version of an apple; a car is a car invariably of it being considered luxury car or not) and so on. The CPI dynamic has real trouble reflecting real consumer spendings.